# Bayesian Statistics for Genetics | 贝叶斯与遗传学

Common sense reduced to computation

- Pierre-Simon, marquis de Laplace (1749–1827)

Inventor of Bayesian inference

## Session 0, Course outline

Bayesian methods

Bayes’ Theorem (Bayes’ Rule)

Bayesian inference

## Session 1, Introductions, motivations for Bayes, overview of Bayesian statistics

CI则是独立的，贝叶斯和频率学派都在用。

In addition to describing random variables, Bayesian statistics uses the ‘language’ of probability to describe what is known about unknown parameters.
Frequentist statistics , e.g. using p-values & confidence intervals, does not quantify what is known about parameters.

Prior distribution: what you know about parameter θ, excluding the information in the data – denoted p(θ)

Likelihood: based on modeling assumptions, how (relatively) likely the data y are if the truth is θ – denoted p(y|θ)

In almost all frequentist inference, confidence intervals take the form θˆ ± 1.96 × stderr where the standard error quantifies the ‘noise’ in some estimate θˆ of parameter θ.

posterior ∝ prior × likelihood，先验、后验和似然函数都是关于参数的分布函数，所以可以统一在一幅图里可视化。

## Session 2, Review of probability, binomial sampling

需要先看一篇文章，PPT里使用该文章作为一个案例。

A powerful and flexible statistical framework for testing hypotheses of allele-specific gene expression from RNA-seq data

PMF和PDF的联系和区别？为什么必须要从这两个角度来衡量随机变量的概率分布？PMF的定义很直觉，就是随机变量每个值对应的各自概率。问题来了，对于连续型随机变量是画不出PMF，所以PDF就诞生了，对于连续型随机变量我们是不会在意一个点的，我们只在乎区间，PDF就是通过积分来求区间概率的。PDF是如何产生的呢？

## Session 3, Binomial sampling continued. R notes [.Rmd, .pdf]

Simply put, to carry out a Bayesian analysis one must specify a likelihood (probability distribution for the data) and a prior (beliefs about the parameters of the model).

1. Estimation: marginal posterior distributions on parameters of interest.
2. Hypothesis Testing: Bayes factors give the evidence in the data with respect to two or more hypotheses, and provide one approach.
3. Prediction: via the predictive distribution.

（1） 写出似然函数
（2） 对似然函数取对数，并整理
（3） 求导数
（4） 解似然方程

## Session 10, Imputation, and software [.R, for DIC/WAIC examples]

Bayesian Statistics for Genetics | 贝叶斯与遗传学

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